reserve i1 for Element of INT;
reserve j1,j2,j3 for Integer;
reserve p,s,k,n for Nat;
reserve x,y,xp,yp for set;
reserve G for Group;
reserve a,b for Element of G;
reserve F for FinSequence of G;
reserve I for FinSequence of INT;

theorem Th19:
  for G being finite Group holds
  ( G is cyclic iff ex a being Element of G st ord a = card G )
proof
  let G be finite Group;
  thus G is cyclic implies ex a being Element of G st ord a = card G
  proof
    reconsider H = the multMagma of G as Group;
    assume G is cyclic;
    then consider a being Element of G such that
A1: the multMagma of G = gr {a};
    take a;
    ord a = card H by A1,Th7;
    hence thesis;
  end;
  given a being Element of G such that
A2: ord a = card G;
  ex a being Element of G st the multMagma of G = gr {a}
  proof
    consider a being Element of G such that
A3: ord a = card G by A2;
    take a;
    card gr {a} = card G by A3,Th7;
    hence thesis by GROUP_2:73;
  end;
  hence thesis;
end;
